Soft spots on the edge of the covered patio roof. I noticed the guttering is pulling away there so I peered in?

I have a couple of soft spots on the edge of the covered patio. The guttering was pulling loose right at these spots and the fascia board and wood behind it are rotted. Obviously I need to replace it, but is it possible to pull up the shingles from the outside edge without damaging them? How would you proceed? The roof is about 13 years old. I can't afford to pay anyone to do this, so I need to do it myself.